calculusgeometryoptimization
Im currently learning how to maximize areas.
theres a question that I'm stuck on
Find the largest trapezoid that can be inscribed in a circle of radius 2 and whose base is the diameter of the circle.
Can anyone help me solve this, or give me pointers.
When you went from $$\sqrt{4-x^2} - {{2x^2 - 4x} \over \sqrt{4-x^2}} = 0$$ to $$ {{4 - x^2 - 2x^2 - 4x} \over \sqrt{4-x^2}} = 0$$ you slipped a sign. The last term in the numerator should be $+4x$ instead of $-4x$.
Never mind. That error just reverses a previous erroneous sign. Your real error is in your first simplification of $A'$. You have factors of $1 \over 2$ and $-2x$. The $2$s should have cancelled, but you instead left out the $1 \over 2$.
Let the radius of the circle be $r$; then the height of the isosceles trapezoid is $2r$, and the length of a lateral side would be $4r$.
The four right triangles with $OB$ and $OC$ as hypotenuses are congruent. The four right triangles with $OA$ and $OD$ as hypotenuses are also congruent. Therefore the lengths marked $x$ are all the same, as are those marked $y$.
The area of a trapezoid is $$\frac{1}{2}h(a+b)$$
Therefore $$S = \frac{1}{2}h(a+b)$$
$$S = \frac{1}{2}(2r)(2x + 2y)$$
$$S = 2r(x + y)$$
But we know $x + y = 4r$. Therefore,
$$S = 2r(4r)$$
$$\therefore r = \sqrt{\frac{S}{8}} = \frac{\sqrt{2}}{4}\sqrt{S} \approx 0.3536 \sqrt{S}$$
Best Answer
One thing to do is to reduce the number of variables. So here is an extended hint. [I've assumed the base is on the diameter, but need not be the whole of it]
For example, there will be two points on the diameter - can you locate those, given that the area is maximum. Suppose you draw a random trapezoid. Try moving one of the points on the diameter and see whether it increases or decreases the area. Where does it need to be to get the biggest area? Do the same for the other point.
Then there are two points on a line parallel to the diameter. Where will these be if the area is a maximum?
Can you see that you need consider only one variable for the position of the line parallel to the diameter, and express your problem in terms of that?
Then you can use calculus on a one variable problem which you know how to tackle.